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Properties

Label 2-2352-7.6-c2-0-61
Degree $2$
Conductor $2352$
Sign $0.654 + 0.755i$
Analytic cond. $64.0873$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 4.38i·5-s − 2.99·9-s + 19.5·11-s − 6.11i·13-s + 7.59·15-s + 8.76i·17-s − 30.3i·19-s + 24·23-s + 5.78·25-s − 5.19i·27-s + 13.5·29-s + 28.0i·31-s + 33.9i·33-s − 48.5·37-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.876i·5-s − 0.333·9-s + 1.78·11-s − 0.470i·13-s + 0.506·15-s + 0.515i·17-s − 1.59i·19-s + 1.04·23-s + 0.231·25-s − 0.192i·27-s + 0.468·29-s + 0.904i·31-s + 1.02i·33-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(64.0873\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1),\ 0.654 + 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.238758173\)
\(L(\frac12)\) \(\approx\) \(2.238758173\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 4.38iT - 25T^{2} \)
11 \( 1 - 19.5T + 121T^{2} \)
13 \( 1 + 6.11iT - 169T^{2} \)
17 \( 1 - 8.76iT - 289T^{2} \)
19 \( 1 + 30.3iT - 361T^{2} \)
23 \( 1 - 24T + 529T^{2} \)
29 \( 1 - 13.5T + 841T^{2} \)
31 \( 1 - 28.0iT - 961T^{2} \)
37 \( 1 + 48.5T + 1.36e3T^{2} \)
41 \( 1 - 7.14iT - 1.68e3T^{2} \)
43 \( 1 + 53.7T + 1.84e3T^{2} \)
47 \( 1 + 39.9iT - 2.20e3T^{2} \)
53 \( 1 + 61.5T + 2.80e3T^{2} \)
59 \( 1 + 77.1iT - 3.48e3T^{2} \)
61 \( 1 + 0.431iT - 3.72e3T^{2} \)
67 \( 1 - 56.9T + 4.48e3T^{2} \)
71 \( 1 - 123.T + 5.04e3T^{2} \)
73 \( 1 - 35.8iT - 5.32e3T^{2} \)
79 \( 1 - 52.1T + 6.24e3T^{2} \)
83 \( 1 - 136. iT - 6.88e3T^{2} \)
89 \( 1 + 15.2iT - 7.92e3T^{2} \)
97 \( 1 + 34.1iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.764591997471838479261155354226, −8.314956737330268374685140935889, −6.88214688215252893418553393709, −6.56111094057089963473172868096, −5.18886114873155789990219849164, −4.86957629712444937724315877341, −3.85895733242819000832047960484, −3.08717856019047749180231636666, −1.60693560935444820782152954596, −0.62987352809032726427524837208, 1.09235107499025842299297622627, 1.98840979755794598100597534865, 3.18907588121943278356054450798, 3.84978226035777102609205426336, 4.97171627564362993168882002685, 6.14734372291068335904887733186, 6.60307637554926958635121863762, 7.19078742882202897060386778203, 8.058236424972238995414268573245, 8.954738504059058878922471800387

Graph of the $Z$-function along the critical line